Verify the Equality: (14+a)(a-2) = -2a²+12a-28

Q: How do I expand (14+a)(a-2) correctly?

Use the distributive property: multiply each term in the first parentheses by each term in the second. So: $ 14 \cdot a + 14 \cdot (-2) + a \cdot a + a \cdot (-2) = 14a - 28 + a^2 - 2a $

Q: What does 'combine like terms' mean?

Like terms have the same variable and exponent. In $ 14a - 28 + a^2 - 2a $, combine 14a and -2a to get 12a, leaving $ a^2 + 12a - 28 $.

Q: Why is the coefficient of a² important?

The coefficient tells you how many times $ a^2 $ appears. Our expansion gives +1 as the coefficient, but the given expression has -2. Since +1 ≠ -2, the equality is false!

Q: Could the other terms also be wrong?

Good thinking! Always check all terms. In this problem, the $ 12a $ and $ -28 $ terms match perfectly. Only the $ a^2 $ coefficient is wrong.

Q: How can I avoid making expansion mistakes?

Work step-by-step: (1) Write out all four multiplication terms, (2) Calculate each one carefully, (3) Combine like terms, (4) Compare each coefficient to the given expression.

Polynomial Expansion with Verification Techniques

Is equality correct?

$(14+a)(a-2)=-2a^2+12a-28$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Are the expressions equal?

00:13 Let's open the parentheses carefully and multiply each factor by every other factor.

00:29 Now, let's calculate the products, step by step.

00:46 Let's group the similar factors together.

00:55 Now, compare the terms of both expressions.

01:01 We can see the expressions are not equal because of the different coefficient of A squared.

01:07 And that's the solution to our question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Is equality correct?

$(14+a)(a-2)=-2a^2+12a-28$

Step-by-step solution

To solve this problem, let's scrutinize the expression $(14+a)(a-2)$ by expanding it:

Distribute each term in the first parenthesis with each term in the second parenthesis:
$(14+a)(a-2) = 14 \cdot a + 14 \cdot (-2) + a \cdot a + a \cdot (-2)$
Simplify each part: $14a - 28 + a^2 - 2a$
Combine like terms: $a^2 + 12a - 28$

Now, compare this expanded and simplified expression $a^2 + 12a - 28$ to the given expression on the right-hand side, $-2a^2 + 12a - 28$ .

Observe that the coefficient of $a^2$ is $1$ in our expansion but $-2$ in the right-hand side expression.

Therefore, the equality is incorrect due to the differing coefficients of $a^2$ in the expressions.

Hence, the correct choice is: No, due to the coefficient of $a^2$ .

Final Answer

No, due to the coefficient of $a^2$

Key Points to Remember

Essential concepts to master this topic

Expansion: Use distributive property to multiply each term systematically
Technique: (14+a)(a-2) = 14a - 28 + a² - 2a = a² + 12a - 28
Check: Compare coefficients term by term: a² vs -2a², +12a vs +12a, -28 vs -28 ✓

Common Mistakes

Avoid these frequent errors

Accepting expressions without expanding and comparing
Don't assume polynomial expressions are equal without expanding = missing coefficient errors! This leads to accepting wrong equations as true. Always expand the left side completely and compare each coefficient.

Practice Quiz

Test your knowledge with interactive questions

$$ (x+y)(x-y)= $$

FAQ

Everything you need to know about this question

How do I expand (14+a)(a-2) correctly?

Use the distributive property: multiply each term in the first parentheses by each term in the second. So: $14 \cdot a + 14 \cdot (-2) + a \cdot a + a \cdot (-2) = 14a - 28 + a^2 - 2a$

What does 'combine like terms' mean?

Like terms have the same variable and exponent. In $14a - 28 + a^2 - 2a$ , combine 14a and -2a to get 12a, leaving $a^2 + 12a - 28$ .

Why is the coefficient of a² important?

The coefficient tells you how many times $a^2$ appears. Our expansion gives +1 as the coefficient, but the given expression has -2. Since +1 ≠ -2, the equality is false!

Could the other terms also be wrong?

Good thinking! Always check all terms. In this problem, the $12a$ and $-28$ terms match perfectly. Only the $a^2$ coefficient is wrong.

How can I avoid making expansion mistakes?

Work step-by-step: (1) Write out all four multiplication terms, (2) Calculate each one carefully, (3) Combine like terms, (4) Compare each coefficient to the given expression.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Algebraic Technique questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime